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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I

Authors: Douglas N. Hoover and Edwin Perkins
Journal: Trans. Amer. Math. Soc. 275 (1983), 1-36
MSC: Primary 60H10; Secondary 03H05
Part II: Trans. Amer. Math. Soc. (1) (1983), 37-58
MathSciNet review: 678335
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Abstract: R. M. Anderson has developed a nonstandard approach to Itô integration in which the Itô integral is interpreted as an internal Riemann-Stieltjes sum. In this paper we extend this approach to integration with respect to semimartingales. Lifting and pushing down theorems are proved for local martingales, semimartingales and other right-continuous processes on a Loeb space.

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Keywords: Stochastic integration, local martingale, semimartingale, quadratic variation, Skorokhod topology, nonstandard analysis
Article copyright: © Copyright 1983 American Mathematical Society